Mathematicians used the "naive" concept of a set as "a collection of objects" for thousands of years, and it led to a major crisis in the foundations of mathematics that was only resolved by establishing a rigorous system of axiom's that define what a set is.
So I disagree that a "set is more than a collection", with the qualification that the concept of a "collection" that is not associated with any axioms whatsoever, is meaningless. Why do I say "meaningless"? Because this "naive" concept of a collection leads to a number of logical paradox's that can only be resolved by using a system of axioms to define these collections.
So it's true that you can define sets by various axiom systems that are not
the same, but these systems all exist under the umbrella of "set theory".
Using the "naive" concept of a collection is fair enough, since we all "know what you mean", but only as long as your point isn't that category theory is more fundamental than set theory.
Actually, it didn't resolve the crisis. In fact, categories, computability theory, and most mathematical findings of the last 150 years are a direct result of stepping around the crisis, namely Russell's paradox.
My point is not that category theory is more fundamental than set theory (though set theory can be described in terms of categories), but rather that it isn't derivative. You don't need the formalisms of set theory to have category theory.
So I disagree that a "set is more than a collection", with the qualification that the concept of a "collection" that is not associated with any axioms whatsoever, is meaningless. Why do I say "meaningless"? Because this "naive" concept of a collection leads to a number of logical paradox's that can only be resolved by using a system of axioms to define these collections.
So it's true that you can define sets by various axiom systems that are not the same, but these systems all exist under the umbrella of "set theory".
Using the "naive" concept of a collection is fair enough, since we all "know what you mean", but only as long as your point isn't that category theory is more fundamental than set theory.